The fact that there is still ambiguity to use the total law of probability in quantum theory though it has been long recognized it is not upheld in quantum mechanics shows the needs to provide a rigorous formulation of the law of total probability in quantum theory, and to clarify under what conditions it holds true. And this is indeed the motivation of the present work. The formulation of the law of total probability depends on a clear definition of conditional probability in quantum theory. There is already extensive research literature on how the conditional probability is defined [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}. However, these formulations are either based on projection measurements, or only considering simultaneous measurements with commutative operators. In this work we will extend a two-time conditional probability formulation from projection measurement to more generic POVM measurements. Generalization for POVM measurement is needed because some of the no-go theorems choose POVM operators in their proofs. We then give several sufficient conditions for the law of total probability to become true. The theory is applied to analyze several no-go theorems related to the extended Wigner's friend scenario. Logical loopholes are shown in these no-go theorems because their proofs rely on the law of total probability one way or another, but the conditions to validate the law are not met. Thus, these no-go theorems do not really prove the results they expect, such as “measured facts are observer-dependent". Instead, they just indirectly confirm that the law of total probability doesn't not hold in quantum theory.
